In the cyclic quadrilateral ABCD AB=7,BC=8,CD=8,DA=15. Calculate the angle ADC and the length ofAC.


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Question Number 19709 by NECC last updated on 14/Aug/17

$I n t h e c y c l i c q u a d r i l a t e r a l A B C D$ $A B = 7, B C = 8, C D = 8, D A = 15 .$ $C a l c u l a t e t h e a n g l e A D C a n d$ $t h e l e n g t h o f A C .$
	Answered by Tinkutara last updated on 15/Aug/17

	Commented by Tinkutara last updated on 15/Aug/17

	$∠ B = π - ∠ D$ $\Rightarrow \cos B + \cos D = 0$ $\frac{7^{2} + 8^{2} - {A C}^{2}}{2 \times 7 \times 8} + \frac{15^{2} + 8^{2} - {A C}^{2}}{2 \times 15 \times 8} = 0$ $\frac{113 - {A C}^{2}}{7} + \frac{289 - {A C}^{2}}{15} = 0$ $3718 = 22 {A C}^{2}$ $A C = 13$ $\cos ∠ D = \frac{15^{2} + 8^{2} - 13^{2}}{2 \times 15 \times 8} = \frac{1}{2}$ $\Rightarrow ∠ A D C = 60 °$
	Commented by NECC last updated on 15/Aug/17

	$t h a n k y o u s i r .$
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